Commuting Solutions of a Quadratic Matrix Equation for Nilpotent Matrices

2018 ◽  
Vol 25 (01) ◽  
pp. 31-44 ◽  
Author(s):  
Qixiang Dong ◽  
Jiu Ding ◽  
Qianglian Huang
Keyword(s):  
Matrix Equation ◽  
Jordan Block ◽  
Nilpotent Matrix ◽  
Quadratic Matrix Equation ◽  
Nilpotent Matrices ◽  
Quadratic Matrix ◽  
Special Case

We solve the quadratic matrix equation AXA = XAX with a given nilpotent matrix A, to find all commuting solutions. We first provide a key lemma, and consider the special case that A has only one Jordan block to motivate the idea for the general case. Our main result gives the structure of all the commuting solutions of the equation with an arbitrary nilpotent matrix.

Newton-Shamanskii Method for a Quadratic Matrix Equation Arising in Quasi-Birth-Death Problems

2014 ◽  
Vol 4 (4) ◽  
pp. 386-395
Author(s):  
Pei-Chang Guo
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Matrix Equation ◽  
Nonnegative Solution ◽  
Minimal Nonnegative Solution ◽  
The Minimal Nonnegative Solution ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix ◽  
Birth Death

AbstractIn order to determine the stationary distribution for discrete time quasi-birth-death Markov chains, it is necessary to find the minimal nonnegative solution of a quadratic matrix equation. The Newton-Shamanskii method is applied to solve this equation, and the sequence of matrices produced is monotonically increasing and converges to its minimal nonnegative solution. Numerical results illustrate the effectiveness of this procedure.

scholarly journals On the Yang-Baxter-like matrix equation for rank-two matrices

2017 ◽  
Vol 15 (1) ◽  
pp. 340-353 ◽  
Author(s):  
Duanmei Zhou ◽  
Guoliang Chen ◽  
Jiu Ding
Keyword(s):  
Matrix Equation ◽  
Full Column Rank ◽  
Complex Matrices ◽  
The Matrix ◽  
Quadratic Matrix Equation ◽  
Quadratic Matrix

Abstract Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.

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